Divide the following complex numbers: $\dfrac{6(\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi))}{\cos(\frac{23}{12}\pi) + i \sin(\frac{23}{12}\pi)}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Solution: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $6(\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi))$ ) has angle $\frac{7}{4}\pi$ and radius 6. The second number ( $\cos(\frac{23}{12}\pi) + i \sin(\frac{23}{12}\pi)$ ) has angle $\frac{23}{12}\pi$ and radius 1. The radius of the result will be $\frac{6}{1}$ , which is 6. The difference of the angles is $\frac{7}{4}\pi - \frac{23}{12}\pi = -\frac{1}{6}\pi$ The angle $-\frac{1}{6}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{1}{6}\pi + 2 \pi = \frac{11}{6}\pi$ The radius of the result is $6$ and the angle of the result is $\frac{11}{6}\pi$.